TY - JOUR

T1 - Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains

AU - Buckmaster, T.

AU - Germain, P.

AU - Hani, Z.

AU - Shatah, J.

N1 - Funding Information:
TB was supported by National Science Foundation Grant DMS-1600868. PG was supported by National Science Foundation Grant DMS-1301380. JS was supported by National Science Foundation Grant DMS-1363013. ZH was supported by National Science Foundation Grant DMS-1600561, a Sloan Fellowship, and a startup fund from the Georgia Institute of Technology.

PY - 2018/7

Y1 - 2018/7

N2 - We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]d with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size ɛ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of “wave turbulence.” Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small ɛ over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝd. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.

AB - We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]d with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size ɛ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of “wave turbulence.” Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small ɛ over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝd. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.

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U2 - 10.1002/cpa.21749

DO - 10.1002/cpa.21749

M3 - Article

AN - SCOPUS:85044451082

SN - 0010-3640

VL - 71

SP - 1407

EP - 1460

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 7

ER -